Optimization of Large-scale Industrial Value Chains

ABSTRACT

Embodiments control industrial supply chains. An embodiment controls a supply chain formed of multiple nodes by obtaining an input-output model for each node. In response, for each node in the supply chain an equation-oriented model is generated using the obtained input-output model corresponding to the node. The generated equation-oriented models of the multiple nodes are integrated with a linking structure to form an optimization model of the supply chain. The optimization model of the supply chain includes a plurality of variables, e.g., interface variables indicating relationships between the generated equation-oriented models for each node in the supply chain. To continue, the optimization model of the supply chain is solved using a categorization of each of the plurality of variables to determine a value for at least one variable of the plurality. In turn, the method outputs a signal indicating the determined value.

RELATED APPLICATION

This application claims the benefit of U.S. Provisional Application No.63/182,124, filed on Apr. 30, 2021. The entire teachings of the aboveapplication are incorporated herein by reference.

BACKGROUND

There is a trend in the petroleum and chemicals industries toward moresupply chain integration and optimization. It is expected that oilrefineries will be more tightly integrated with base petrochemicalplants, chemical derivatives plants, and downstream polymer producingfacilities.

SUMMARY

The trend toward supply chain integration and value chain optimizationis creating a need for integrated software solutions that supportintegrated decision-making. Currently, nodes, e.g., separate locationsin a supply chain, in the downstream petroleum and chemical supply chainare operated mostly independently from each other within local decisionenvelopes. The interfaces between nodes, such as material and energystreams exchanged between oil refinery and related downstream chemicalplants, are managed in sub-optimal ways using heuristic and myopicbusiness rules.

Embodiments solve these problems and provide improved functionality forcontrolling supply chains and the nodes therein. It is noted that whileexample embodiments are described herein in relation to petroleum andchemicals industries, embodiments are not so limited and may be used tocontrol any supply chains.

An example embodiment is directed to a method for controlling anindustrial supply chain. The method obtains an input-output model foreach node in a supply chain, formed of multiple nodes. In response, anequation-oriented model is generated for each node in the supply chainusing the obtained input-output model corresponding to the node. Tocontinue, such an embodiment integrates the generated equation-orientedmodels of the multiple nodes with a linking structure to form anoptimization model of the supply chain. The optimization model of thesupply chain includes a plurality of variables, e.g., interfacevariables indicating relationships between the generatedequation-oriented models for each node in the supply chain. To continue,the optimization model of the supply chain is solved using acategorization of each of the plurality of variables to determine avalue for at least one variable of the plurality. The method outputs asignal indicating the determined value. According to an embodiment, thisoutputting enables controlling a given node of the multiple nodes in thesupply chain in accordance with the determined value.

According to another embodiment, each obtained input-output modelincludes one or more inputs and one or more outputs. In such anembodiment the one or more inputs are configured to be manipulated tooptimize the one or more outputs with respect to an objective functionor performance indicator. Example objective functions include maximizingprofit, maximizing revenue, minimizing cost, maximizing resiliency,minimizing risk, and maximizing customer satisfaction, amongst others.

Embodiments can implement one or more functionalities to generateequation-oriented models. An embodiment generates an equation orientedmodel by performing at least one of: (i) processing a given input-outputmodel to generate a matrix indicating logistic, economic, andoperational constraints for a supply chain node corresponding to thegiven input-output model, (ii) using a given input-output model andfitting parameters to a first-principles engineering model, and (iii)processing a given input-output model using at least one of statistics,machine learning, and artificial intelligence.

In an embodiment, the linking structure includes at least one of:mathematical variables and equations connecting the generatedequation-oriented models. As part of solving the optimization model, anembodiment assigns each of the plurality variables to one of a pluralityof categories. Example categories, according to an embodiment, include:(i) variables connecting nodes in the supply chain, (ii) variables in anobjective function, (iii) variables in linear constraints in theoptimization model, and (iv) variables exclusively in nonlinearconstraints.

Another embodiment solves the optimization model by grouping linearconstraints in the optimization model and grouping nonlinear constraintsin the optimization model. Such an embodiment then removes the groupednonlinear constraints from the optimization model. The nonlinearconstraints are removed by: (i) for each nonlinear constraint, creatinga corresponding linear constraint by substituting each variablecategorized as exclusively in the nonlinear constraint, with a lower andupper bound of the variable and (ii) replacing each nonlinear constraintin the optimization model with the created corresponding linearconstraint. In turn, the optimization model with the removed nonlinearconstraints is solved to determine optimal values for variablescategorized as connecting supply chain nodes or variables that appear inan objective function. These determined optimal values are passed backto given equation-oriented models that contain the nonlinear constraintsand the given equation oriented models are solved. If anequation-oriented model is feasible the solving converges and if anequation oriented model is infeasible, the solving does not converge. Inresponse to an equation-oriented model not converging, an embodimentcreates a cut constraint in terms of variables categorized as at leastone of: connecting supply chain nodes and appearing in an objectivefunction, and adds the cut constraint to the optimization model. Thisprocedure is repeated, i.e., iterated until solving eachequation-oriented model converges.

Yet another embodiment obtains the outputted signal and determines acontrol parameter of the given node using the determined value, from theobtained signal, in a given equation-oriented model indicating behaviorof the given node. Such a method operates the given node in accordancewith the determined control parameter.

Nodes may include any supply chain elements known to those of skill inthe art. For example, in an embodiment the nodes in the supply chaininclude: a refinery node, a petrochemical plant, i.e., cracker, node,and a polymer plant node.

Another embodiment is directed to a computer system for controlling asupply chain. The computer system includes a processor and a memory withcomputer code instructions stored thereon. In such an embodiment, theprocessor and the memory, with the computer code instructions, areconfigured to cause the system to control a supply chain in accordancewith any embodiment or combination of embodiments described herein.

Yet another embodiment is directed to a computer program product forcontrolling a supply chain. The computer program product comprises oneor more non-transitory computer-readable storage devices and programinstructions stored on at least one of the one or more storage devices.The program instructions, when loaded and executed by a processor, causean apparatus associated with the processor to control a supply chain asdescribed herein.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing will be apparent from the following more particulardescription of example embodiments, as illustrated in the accompanyingdrawings in which like reference characters refer to the same partsthroughout the different views. The drawings are not necessarily toscale, emphasis instead being placed upon illustrating embodiments.

FIG. 1 is a flowchart of a method of controlling a supply chainaccording to an embodiment.

FIG. 2 is a schematic view of an embodiment of the present invention.

FIG. 3 is a simplified block diagram of a supply chain, i.e., valuechain, that may be optimized utilizing embodiments.

FIG. 4 is block diagram illustrating nodes in a supply chain and therelationship between the nodes, that may be optimized using embodiments.

FIG. 5 is a plot showing result convergence of an embodiment.

FIG. 6 is a simplified block diagram of a system that may be optimizedusing embodiments.

FIG. 7 are tables of results of operational conditions of the systemillustrated in FIG. 6, determined using existing methods.

FIG. 8 are tables of results of operational conditions of the systemillustrated in FIG. 6, determined using embodiments.

FIG. 9 is a schematic view of a computer network environment in whichembodiments of the present invention may be deployed.

FIG. 10 is a block diagram of a computer node in the network of FIG. 9.

DETAILED DESCRIPTION

A description of example embodiments follows.

Embodiments provide improved functionality for controlling supply chainsand enable integrated-decision making across the supply chain so as tomaximize overall value-creation. Embodiments include a framework,including an application architecture, algorithms, and tools, foroptimizing value creation across the entire supply chain. In the longterm, embodiments of the present invention enable the smooth transitionfrom using crude oil feedstocks for fuel production to producing othervaluable chemicals and ultimately replacing crude oil with economicaland sustainable bio-based feedstocks.

Example Method

FIG. 1 is a flowchart of an example method 100 for controlling a supplychain according to an embodiment. In an embodiment, the supply chain isformed of multiple nodes. For instance, in an embodiment the nodes inthe supply chain include: a refinery node, a petrochemical plant node,and a polymer plant node. Moreover, it is noted that embodiments are notlimited to the foregoing nodes, and nodes may include any supply chainelements known to those of skill in the art. In one such embodiment,each node represents a distinct physical location.

The method 100 is computer implemented. As such, the method 100 can beimplemented using any computing device or combination of computingdevices known in the art. Moreover, embodiments of the method 100 can beimplemented in a control tower of a supply chain to monitor and controlelements of the supply chain.

The method 100 begins at step 101 by obtaining an input-output model foreach node in a supply chain, formed of multiple nodes. The method 100 iscomputer implemented and, as such, input-output models may be obtainedat step 101 from any point(s), e.g., storage memory, communicativelycoupled to one or more computing devices implementing the method 100 orpoint(s) that are capable of being communicatively coupled to the one ormore computing devices implementing the method 100. According to anembodiment, each input-output model obtained at step 101 includes one ormore inputs and one or more outputs. An example of an input-output modelobtained at step 101 is the model of a production plant where the inputsare raw materials and the outputs are the finished products. Yet anotherexample is a set of equations corresponding to a chemical process wherethe inputs are the reactants, and the outputs are the products of thereaction. In such an embodiment, the one or more inputs are configuredto be manipulated to optimize the one or more outputs with respect to anobjective function or performance indicator. Thus, in such anembodiment, an objective function is an element of the input outputmodel. Example objective functions include maximizing profit, maximizingrevenue, minimizing cost, maximizing resiliency, minimizing risk, andmaximizing customer satisfaction, amongst other examples.

Also at step 101, an equation-oriented model is generated for each nodeusing the obtained input-output model corresponding to the node.Embodiments of the method 100 can implement one or more functionalitiesto generate equation-oriented models at step 101. An embodimentgenerates an equation oriented model by performing at least one of: (i)processing a given input-output model to generate a matrix indicatingone or more of logistic, economic, and operational constraints for asupply chain node corresponding to the given input-output model (e.g.,material balances, inventory constraints, capacity constraints, andsupply and demand parameters in a manufacturing site), (ii) using agiven input-output model and fitting parameters to a first-principlesengineering model (e.g., kinetic equations, and heat and mass transferequations in a chemical reactor), and (iii) processing a giveninput-output model using at least one of statistics, machine learning,and artificial intelligence (e.g., a surrogate model of a pharmaceuticalproduction process obtained by statistical regression on historicaldata).

Returning to FIG. 1, at step 102 the generated equation-oriented modelsof the multiple nodes are integrated with a linking structure to form anoptimization model of the supply chain. The optimization model of thesupply chain includes a plurality of variables, e.g., interfacevariables indicating relationships between the generatedequation-oriented models for each node in the supply chain. Further,according to an embodiment, the linking structure includes at least oneof: mathematical variables and equations connecting the generatedequation oriented models. To illustrate linking structures, consider anexample supply chain comprising nodes representing basic petrochemicalprocesses and other nodes representing polymerization plants. In such anillustrative example, the linking structure is made of transferequations of monomers produced by the basic petrochemical nodes andconsumed by polymerization nodes.

To continue, at step 103, the optimization model of the supply chain issolved using a categorization of each of the plurality of variables soas to determine a value for at least one variable of the plurality. Aspart of solving the optimization model at step 103, an embodimentassigns each of the plurality variables to one of a plurality ofcategories. Example categories, according to an embodiment include: (i)variables connecting nodes in the supply chain, (ii) variables in anobjective function, (iii) variables in linear constraints in theoptimization model, and (iv) variables exclusively in nonlinearconstraints.

Another embodiment of the method 100 solves the optimization model atstep 103 by grouping linear constraints in the optimization model andgrouping nonlinear constraints in the optimization model. Such anembodiment then removes the grouped nonlinear constraints from theoptimization model. The nonlinear constraints are removed by: (i) foreach nonlinear constraint, creating a corresponding linear constraint bysubstituting each variable categorized as exclusively in the nonlinearconstraint, with a lower and upper bound of the variable and (ii)replacing each nonlinear constraint in the optimization model with thecreated corresponding linear constraint. In turn, the optimization modelwith the removed nonlinear constraints is solved to determine optimalvalues for variables categorized as connecting supply chain nodes orvariables that appear in an objective function that is contained in theoptimization model, e.g., in an equation-oriented model of theoptimization model. These determined optimal values are passed back togiven equation-oriented models that contain the nonlinear constraintsand the given equation oriented models are solved. If anequation-oriented model is feasible the solving converges and if anequation oriented model is infeasible, the solving does not converge. Inresponse to an equation-oriented model not converging, an embodimentcreates a cut constraint in terms of variables categorized as at leastone of: connecting supply chain nodes and appearing in an objectivefunction and adds the cut constraint to the optimization model. Thisprocedure is repeated, i.e., iterated, until solving eachequation-oriented model converges.

In turn, the method 100, at step 104 outputs a signal indicating thedetermined value, i.e., the value determined at step 103 for the atleast one variable. According to an embodiment, this outputting enablescontrolling a given node of the multiple nodes in the supply chain inaccordance with the determined value. In an embodiment, the outputting104 provides the signal to a given node in the supply chain via a knowncommunication methodology, e.g., one or more network connections. Such anode is then controlled, e.g., automatically, through one or morecomputing devices implementing instructions in response to the receivedsignal. Amongst other examples, the node may be controlled to produce anamount of a product indicated by the signal.

Embodiments of the method 100 can react to disruptions in the supplychain by providing modified recommendations for operating the nodes inthe supply chain given the disruptions. For example, if a disruption,e.g., zero output, occurs in a node corresponding to a basicpetrochemical plant, its impact will propagate to a polymerization plantdownstream of the basic petrochemical plant, via the connectingstructure. An embodiment, e.g., the method 100, will identify that zerooutput is coming from the node corresponding to the basic petrochemicalplant and the method will automatically determine that thepolymerization plant should receive input from an alternative sourcelinked to the polymerization plant in the optimized model. This inputcan be sent to the polymerization plant in the real-world so that thepolymerization plant will obtain raw materials from the alternativesource, e.g., an external market, connected to the polymerization plantin the optimization model. The polymerization plant is able to react,because the polymerization plant is represented by an input-output modelthat can be optimized for different values of the inputs.

Another embodiment of the method 100 obtains the outputted 104 signaland determines a control parameter of the given node by using thedetermined value (from the signal) in a given equation-oriented modelthat indicates behavior of the given node. Such a method operates thegiven node in accordance with the determined control parameter.

To illustrate the method 100, consider an example where the supply chainincludes a refinery and a petrochemical plant. In such an example, atstep 101 an input-output model is obtained for the refinery and an inputoutput model is obtained for the petrochemical plant. The model for therefinery and the petrochemical can be obtained from the mixed-integerlinear or nonlinear programming models typically used for productionplanning in such facilities, Equation oriented models are then generatedfor each of the refinery and petrochemical plant. At step 102, theseequation-oriented models are linked by defining a constraint whichrequires a chemical output of the refinery to be an input quantity tothe petrochemical plant. Such inputs are determined by the chemicalprocesses in the refinery and the petrochemical plant. For instance,petrochemical plants consume Naphtha, which is a common product ofrefinery processes. Therefore, this material becomes an output for therefinery and an input for the petrochemical plant. This linked model isthen solved to determine an optimized amount of the chemical to beproduced by the refinery. Because the models of the refinery andpetrochemical plant were linked and solved together, the optimizedoutput of the refinery is an optimized quantity of input to thepetrochemical plant given the constraint (output of refinery is input tothe petrochemical plant). In real-world operation, the refinery is thencontrolled to produce the determined optimized output.

Framework for Value Chain Optimization

FIG. 2 depicts the main elements of a framework 220, according to anembodiment, and outlines how these elements interact. Embodiments, e.g.,the framework 220 of FIG. 2, can be implemented using computing devicesconfigured with program instructions to carry out the variousfunctionalities described herein. Moreover, such implementations mayleverage any number and variety of data structures for storing andholding the models and variables used within the framework 220.

The framework 220 utilizes input-output models 221 a-n (generallyreferred to as 221) for nodes in the supply chain where the inputs forthe models 221 a-n can be manipulated to optimize the output withrespect to an objective function or indicator of performance. Accordingto an embodiment, the input-output models 221 a-n are genericinput-output models. Moreover, in an embodiment, the models 221 a-ncorrespond to each node, e.g., physical location, in the supply chain.

Generation of Representation of Each Model in the Supply Chain

The framework 220 includes a math model constructor element 222. Themath model constructor 220 is configured to programmatically create anequation-oriented representation 223 a-n of each node model 221 a-n. Inembodiments, the math model constructor 222 can have differentimplementations. For instance, the math model constructor 222 canimplement a mathematical matrix generator that constructs at least oneof logistic, economic, and operational constraints usually encounteredin a supply chain node. For example, the constructor can take data froma database and construct material balance and property balanceequations, along with supply and demand constraints. These wouldrepresent the equation-oriented model of a node in the supply chain. Inanother embodiment, the math model constructor 222 implements analgorithm that fits parameters to a first-principles engineering model(e.g., solving parameter estimation problems where process data is usedto fit kinetic reaction constants) and, in yet another embodiment, themath model constructor 222 implements a data-driven algorithm thatcreates an equation-oriented model 223 a-n out of a black-box,input-output model 221 a-n, using statistics, machine learning, and/orArtificial Intelligence techniques. An example of such functionalityincludes the generation of a surrogate model made of linear, polynomial,and exponential equations fitted to historical data. Those equationscontain the inputs as independent variables, and the outputs asdependent variables

Integration of Individual Mathematical Representations into OneIntegrated Value Chain Optimization Model

The framework 220 integrates the equation oriented models 223 a-n usinglinking structures 224 a-n to create the single optimization model 225.In an embodiment, the linking structures 224 a-n are based on user inputindicating, amongst other examples, elements, decisions, variables, andequations, that connect the different nodes of the supply chain. In anembodiment, a user provides the linking structures 224 a-n via a userinterface. In the framework 220, each of the equation-oriented models223 a-n built by the math model constructor 222 are incorporated into alarge-scale optimization model 225 and integrated through the linkingstructures 224 a-n. An example linking structure 224 a-n is an equationthat sets the production of ethylene in an Olefins Plant (which isrepresented by a node and corresponding equation-oriented model e.g.,223 a) to be equal to the consumption of ethylene in a PolymerizationPlant (which is represented by a node and correspondingequation-oriented model e.g., 223 b). The resulting model 225 may havethe following characteristics: (i) millions of continuous variables,(ii) millions of equations and inequality constraints containing bothlinear and nonlinear terms, and (iii) hundreds of thousands of discretevariables.

Solving Large-Scale Previously Unsolvable Value Chain OptimizationModels

In an embodiment, the framework 220 groups constraints in theoptimization model 225 as linear or nonlinear. An example of a group oflinear constraints, according to an embodiment, is the set materialbalance equation in a refinery. An example of a group of nonlinearconstraints, in an embodiment, is the set of property balances thatoccur in pooling or blending of materials in a process plant. Variables,e.g., production of premium gasoline in a refinery, or the vaporpressure of a such a gasoline, of these constraints are assigned atleast one of four categories (category 1, category 2, category 3, andcategory 4). With the exception of category 4, variables can appear inmore than one category. According to an embodiment, the categoriesinclude: Category 1—variables that connect the supply chain nodes (i.e.,variables in the linking structures 224 a-n, such as ethylene exchangedbetween Olefins and Polymerization plants); Category 2—variables thatappear in an objective function of one of the equation oriented models223 a-n, for example, sales of monomer to an external market; Category3—variables that appear in linear constraints, for example, initialinventory of diesel in a refinery; and Category 4—variables that appearin nonlinear constraints exclusively, for example, the Octane Number ofa gasoline calculated using nonlinear empiric equations.

After grouping the constraints and categorizing the variables, a masterproblem is defined by excluding all nonlinear constraints from thelarge-scale, integrated optimization model 225. The nonlinearconstraints are excluded by replacing the nonlinear constraints with newconstraints. These new constraints are created by substituting thevariables in Category 4 for their lower and upper bound. The upper andlower bounds represent a linearized projection of the nonlinearconstraints. These linearized constraints are added to the masterproblem.

In turn, the master problem which is a large-scale linear ormixed-integer linear model, is solved. This solving determines optimalvalues for variables in Categories 1 and 2. These values are passed backto the individual equation-oriented models 223 a-n of nodes in thesupply chain that contain nonlinear constraints.

The individual equation-oriented models 223 a-n with the determinedvalues for variables in categories 1 and 2 are solved. In an embodiment,the individual equation-oriented models 223 a-n are solved using knownmethodologies (e.g., through linear programming). If the problem(solving the equation oriented models 223 a-n) is feasible the algorithmconverges; if the problem is infeasible, a new constraint is writtenexclusively in terms of Category 1 and Category 2 variables and added tothe master problem 225, e.g., as another equation in the master problem225. This constraint is referred to as a “cut” and it can be generatedin several ways. For instance, the constraint can be generated through:(i) an integer cut, (ii) a Benders cut, (iii) a logic-based cut, and(iv) a cut generated through data-driven and/or artificial intelligencemeans.

The master problem is solved iteratively until the individualequation-oriented models 223 a-n are all feasible. For example, aftermaking the cut, the model 225 is updated to include any valuesdetermined by solving the equation-oriented models 223 a-n. If at leastone equation-oriented model 223 a-n does not converge, a cut isgenerated and added to the model 225. This updated model is then solvedas described above by grouping and categorizing variables and thisprocess continues and repeats until each equation-oriented model 223 a-ncan be solved (converges).

Processing Output from the Integrated Value Chain Optimization Model

After obtaining a solution to the large-scale integrated optimizationmodel 225, the optimal values 226 of the variables that connect thedifferent supply chain nodes are passed to the individual models 221a-n. These are the Category 1 variables, which, when fixed in each ofthe models 221 a-n of the supply chain, allow each model 221 a-n to beran independently. Since the integrated variables are solved tooptimality, the individual solutions to each model 221 a-n represent thebest solution for the whole, integrated, supply-chain.

Value Chain Optimization—Illustrative Example

Below, in relation to FIGS. 3-8, an illustrative example and results aredescribed. In this illustrative example, the objective is to define andsolve an integrated oil to chemicals planning problem.

FIG. 3 is a simplified block diagram of a supply, i.e., value, chain330. The value chain 330 includes the margin-driven supply chain node331 and demand-driven supply chain node 332. There is an optimizationboundary 333 that is based on resource allocation 334, i.e., a processwhere the amount of resources assigned (allocated) to a productionprocess is determined by what is most profitable (profit-driven), andresource nomination 335, i.e., required resources predetermined(nominated) by a demand-driven process, between the margin-driven node331 and the demand-driven node 332. In an example, the margin-drivensupply chain node 331 is a refinery that produces olefins and aromaticsand the demand-driven supply chain node 332 creates derivatives andpolymers using products from the margin-driven supply chain node 331.Embodiments can be used to determine an optimized resource allocation334 from the margin-driven node 331 to the demand-driven node 332 and anoptimized resource nomination 335 from the demand-driven node 332 to themargin-drive node 331.

An embodiment determines a single model to maximize value across thevalue chain 330. Such an embodiment aligns incentives (objectivefunctions) for overall value creation. Embodiments streamline tools andwork processes and replace existing heuristic-based methods forcontrolling supply chain nodes with rigorous optimization methods. Thisallows embodiments to optimize complex decisions across the value chain,e.g., crude and feedstock selection and optimization, synergies betweenrefinery and chemical streams disposition and allocation, fuel versuschemical production, buy versus sell versus make along the value chain.

In the example described below, it is assumed that there are standaloneProfit Impact of Market Strategy (PIMS) planning models for refinery andbasic petrochemicals, and Supply Planner (SP) models for polymers andderivatives. It is also assumed that the PIMS and SP models solve in areasonable time and provide quality solutions when ran independently.Further, it is assumed that there are relatively few variables that arecommon to both the SP and PIMS models compared to the total number ofvariables in both models.

Refining/basic petrochemicals and polymer/derivatives are usuallydifferent businesses that use different software solutions. Real-liferefineries and basic petrochemicals models result in very complexNonlinear Programming Problems (NLPs) that are modeled and solved withPIMS solutions, such as those provided by the Applicant. Realisticpolymer plants and their complex distribution networks are modeled aslarge-scale Mixed-integer Linear Programming Problem (MILPs) and solvedusing software tools, such as AspenTech Supply Planner (SP). PIMS isspecialized for NLPs and MINLPs with few binary variables but manybilinear, trilinear, and general nonlinear terms. SP is used forlarge-scale MILP models with thousands of binary variables.

There is a trend towards more integration of refineries, petrochemicalsplants, and polymer facilities. At the software level, this translatesinto an incentive to integrate tools that have been traditionally keptseparate. Since PIMS and SP both generate matrices of constraints thatare solved as optimization problems, they could conceptually beintegrated by adding connecting structure to represent the streamsexchanged between both businesses. This connecting would allow theoptimization of the whole system, in what is referred to herein as“Value Chain Optimization”.

Integrating the mathematical programming models from real-life PIMS andSP models results in large-scale MINLP models. In order to solve thelarge-scale MINLP models within a reasonable computational time (5-10minutes as a rule of thumb), embodiments implement a decompositionmethodology. The description that follows describes an example of suchfunctionality.

FIG. 4 shows a simplified integrated model that has a sectionrepresenting a node 441 that produces monomers from naphtha that wouldtypically be modelled in PIMS and a section representing a node 442 thatproduces polymers from monomers that would be modelled in SP. For thisillustrative example, the problem was generated outside of PIMS and SPby writing the constraints directly in a .mps file (text file) that canbe read by a MINLP solver. However, for illustrative purposes, themonomer/olefins section 441 is referred to herein as the PIMS part ofthe model as if it had been created as a PIMS model. Likewise, thepolymer section 442 is referred to herein as the SP part of the model.

The PIMS section 441 is an NLP that includes pooling of naphtha sourcesN1 443 a and N2 443 b into naphtha pool N 444. This pooling involves thecalculation of property q, which corresponds to the naphtha poolspecific gravity (spg). Naphtha 444 is turned into monomers 446 a and446 b, by plant 445, which can be sold (S variables 447 a and 447 b) ortransferred to polymer production (M^(PIMS) variables 448 a and 448 b).

The SP planner section 442 is an MILP that includes the transformationof monomers 449 a and 449 b into polymers 451 a and 451 b, via processes450 a and 450 b, respectively. A requirement of minimum batch sizes inthe polymerization is modeled using binary variables.

In FIG. 4 each dotted rectangle contains the elements of each separateapplication (node 441 and node 442), and the two equalities 452 a and452 b connecting nodes 441 and 442 represent the connecting structure.In this example, equality 452 a indicates that M^(PIMS) variable 448 ais equal to monomer 449 a and the equality 452 b indicates that M^(PIMS)variable 448 b is equal to monomer 449 b.

Equations (2)-(6) and (13)-(18) represent a simplification of the typeof constraints in a PIMS model 441. This simplified model is referred toherein as “Primary Distribution”. Equations (9)-(12), (19), and (20)represent a simplification of the type of constraints in the SP model442. This simplified model is referred to herein as “SecondaryDistribution”. Equations (7), (8) connect sections 441 and 442, andequation (1) is an integrated objective function.

Profit=9.5S ₁+7.5S ₂+15P ₁+15P ₂−5N ₁−7N ₂  (1)

N=N ₁ +N ₂  (2)

N spg=N ₁0.7+N ₂0.8  (3)

N spg=M ₁ +M ₂  (4)

M ₁ =S ₁ +M ₁ ^(Primary)  (5)

M ₂ =S ₂ M ₂ ^(Primary)  (6)

M ₁ ^(Primary) =M ₁ ^(Secondary)  (7)

M ₂ ^(Primary) =M ₂ ^(Secondary)  (8)

P ₁=0.9M ₁ ^(SP)  (9)

P ₂=0.8M ₂ ^(Secondary)  (10)

y ₁5≤P ₁ ≤y ₁8  (11)

y ₂2≤P ₂ ≤y ₂10^(P)  (12)

spg _(min) ≤spg≤spg _(max)  (13)

N≤20  (14)

N ₁≤8  (15)

N ₂≤13  (16)

S ₁≤7  (17)

S ₂≤5  (18)

P ₁ +P ₂≥12  (19)

y ₁ ,y ₂∈{0,1}  (20)

All variables nonnegative  (21)

If the system 440, governed by the above equations (1)-(21) is solvedwithout decomposition, e.g., using an existing MINLP solver provided bythe Applicant, the optimal value of the objective function, equation(1), is 71.733. In this solution, objective value of 71.733 determinedwithout decomposition, N=20, N1=8, N2=12, S=0, S2=0, P1=8, P2=5.05, M₁^(Primary)=8.89, and M₂ ^(Primary)=6.31.

If variables in equations (7) and (8) are fixed, the PIMS section 441and SP section 442 can be solved independently. An assumption, accordingto an embodiment, is that individual node models, e.g., SecondaryDistribution 442 and Primary Distribution 441 models, are used that canbe solved independently in an acceptable amount of time (5-10 minutesmax). From that perspective, variables M₁ ^(Primary), M₁ ^(Secondary),M₂ ^(Primary), M₂ ^(Secondary) are complicating variables in the sensethat once they are fixed, the problem becomes significantly easier tosolve. Each of the decomposed problems, i.e., the individual problemsfor the nodes 441 and 442, corresponds exactly to an independentSecondary Distribution model and independent Primary Distribution model.

The notion of complicating variables evokes the idea of BendersDecomposition. The problem could alternatively be interpreted as havingcomplicating constrains that would suggest the use of LagrangeanDecomposition. Furthermore, it is known (although it is not evident inthe illustrative example) that a problem with linearized constraints iseasier to solve than the original MINLP. This last notion could lead toa Linear/Nonlinear decomposition strategy.

A methodology implemented by embodiments is a mix of Benders andLinear/Nonlinear approaches. Compared to standard Benders, embodimentscreate a much stronger Master Problem from iteration zero. Compared to apure Linear/Nonlinear approach, embodiments have the strength ofgenerating benders feasibility and optimality cuts to enrich the linearsubproblem.

Decomposition Methodology Applied to the Illustrative Example

Master Problem First Iteration

For the first iteration M₁ ^(Primary), M₁ ^(Secondary), M₂ ^(Primary),M₂ ^(Secondary) are defined as complicating variables in the Benderssense. However, instead of just writing the problem in terms of thosevariables, an embodiment adds all the linear structure of the originalMINLP to strengthen the master problem, i.e., a “linearized” version ofan integrated model, e.g., 225. Such an embodiment goes even further andadds linearized relaxations of the original nonlinear constraints.Embodiments can afford to do this because the resulting master problemis still easy to solve. The resulting master problem is easy to solvebecause such an embodiment starts by assuming that there is an SecondaryDistribution model that is easy to solve by itself, and because thedimensionality of realistic Primary Distribution models (441), such asPIMS models in the industry, is about at least an order of magnitudesmaller than realistic instances of the original Secondary Distributionmodel (442), such as those SP models used in the industry.

The master problem at the first iteration substitutes the nonlinearconstraints N*spg=N₁ 0.7+N₂ 0.8, N*spg=M₁+M₂, and 0.1125<=spg<=1.125 forlinear relaxations thereof. The master problem at this first iterationis governed by the following equations:

Profit≤π_(S,1) S ₁+π_(S,2) S ₂+π_(P,1) P ₁+π_(P,2) P ₂−π_(N,1) N₁−π_(N,2) N ₂  (1)

s.t. (such that)

Equations (2) and (5)-(21) as above.

N spg _(min) ≤N ₁0.7+N ₂0.8  (3a)

N ₁0.7+N ₂0.8≤N spg _(max)  (3b)

N spg _(min) ≤M ₁ +M ₂  (4a)

M ₁ +M ₂ ≤N spg _(max)  (4b)

The linearization strategy used is explained in detail below.

The master problem is an MILP. The master problem is a relaxation of theoriginal problem. In this first iteration the optimal solution of themaster problem is 156.47, N=20, N1=8, N2=12, S1=1.1, S2=0. {circumflexover (P)}₁=8, {circumflex over (P)}₂=10, {circumflex over (M)}₁^(PIMS)=8.89, {circumflex over (M)}₂ ^(PIMS)=12.5 (with hats) are thevalues at the optimal solution of the master. They are fixed in thesubproblem.

Subproblem First Iteration

In the subproblem, the values of the complicating variables are fixedand the resulting problem (with the fixed complicating variables) issolved. The subproblem at the first iteration is governed by theequations that follow.

Profit=π_(S,1) S ₁+π_(S,2) S ₂+π_(P,1)8+π_(P,2)12−π_(N,1) N ₁−π_(N,2) N₂  (1)

N=N ₁ +N ₂  (2)

N spg=N ₁0.7+N ₂0.8  (3)

N spg=M ₁ +M ₂  (4)

M ₁ =S ₁+8.89  (5)

M ₂ =S ₂+12.5  (6)

spg _(min) ≤spg≤spg _(max)  (13)

N≤20  (14)

N ₁≤8  (15)

N ₂≤13  (16)

S ₁≤7  (17)

S ₂≤5  (18)

The subproblem is an NLP. The subproblem is a lower bound of theoriginal problem. In the first iteration the subproblem is infeasible.Therefore, the following feasibility problem is created.

minimize Infeasibility=Σμ_(i)  (1)

N=N ₁ +N ₂+μ₁−μ₂  (2)

N spg=N ₁0.7+N ₂0.8+μ₃−μ₄  (3)

N spg=M ₁ +M ₂+μ₅−μ₆  (4)

M ₁ =S ₁+8.89+μ₇−μ₈  (5)

M ₂ =S ₂+12.5+μ₉−μ₁₀  (6)

spg _(min) ≤spg≤spg _(max)  (13)

N≤20+μ₁₁  (14)

N ₁≤8+μ₁₂  (15)

N ₂≤13+μ₁₃  (16)

S ₁≤7+μ₁₄  (17)

S ₂≤5+μ₁₅  (18)

0≤μ_(i)  (22)

The following Benders Feasibility Cut is created from the solution ofthe feasibility problem:

M ₁ +M ₂≤15.3

This feasibility cut has a nice physical interpretation. In equation (4)the amount of monomer produced is limited by the total naphtha purchasedtimes the naphtha pool spg. The feasibility cut constrains the totalmonomer (M₁+M₂) to be less than the active constraint on naphtha N≤20times the spg calculated by solution to the subproblem (˜0.75).

Second Iteration of Master Problem

For the second iteration of solving the master problem a relaxation ofnon-linear constraints, such as the one described above, is once againimplemented. Further, a new equation derived from the subproblemsolution that improves the master is employed. Specifically, the seconditeration of the master problem now includes constraint (23) below,which is the infeasibility cut derived from the subproblem. For thesecond iteration the following equations apply:

Profit=π_(S,1) S ₁+π_(S,2) S ₂+π_(P,1) P ₁+π_(P,2) P ₂−π_(N,1) N₁−π_(N,2) N ₂  (1)

s.t.

Equations (2) and (5)-(21) as above.

Nq _(min) ≤N ₁ q ₁ +N ₂ q ₂  (3a)

N ₁ q ₁ +N ₂ q ₂ ≤Nq _(max)  (3b)

Nq _(min) ≤M ₁ +M ₂  (4a)

M ₁ +M ₂ ≤Nq _(max)  (4b)

M ₁ +M ₂≤15.3  (23)

The solution to the second iteration of the master problem is 140.9,N=20, N1=8, N2=12, S1=7, S2=0.2, P1=8, P2=5.13, M₁ ^(PIMS)=8.89, and M₂^(PIMS)=6.4. This solution leads to a feasible subproblem solution.

Convergence of the Methodology

Starting on the second iteration the subproblem is feasible. The problemconverges in four iterations. The feasibility cut already described plusthe two optimality cuts below are added to the master problem throughoutthe methodology iterations:

M ₁ +M ₂≤15.3

Profit≤180+15(P ₁ +P ₂)−20(M ₁ ^(PIMS) +M ₂ ^(PIMS))

Profit≤20.4+15(P ₁ +P ₂)−9.5(M ₁ ^(PIMS) +M ₂ ^(PIMS))

The convergence of the methodology is shown in the plot 550 of FIG. 5where the objective value 551 is plotted versus the iterations 552. Theplot 550 includes series for the subproblem 553, master problem 554, andfull space (FS) solution 555. As noted, the plot 550 shows that themaster problem 554 and subproblem 553 converge at the fourth iteration552.

Formal Definition of Decomposition Methodology

1. Define Integrated Planning Problem (IP):

$\begin{matrix}{\max\limits_{x,y,z}\{ {{f_{Primary}( {x,y} )} + {f_{Secondary}( {z,y} )}} \}} & (1)\end{matrix}$ $\begin{matrix}{{g( {x,y} )} \leq \begin{matrix}0 & ( {{all}{the}{original}{Primary}{Distribution}{inequalities}} )\end{matrix}} & (2)\end{matrix}$ $\begin{matrix}{{h( {x,y} )} = \begin{matrix}0 & ( {{all}{the}{original}{Primary}{Distribution}{equalities}} )\end{matrix}} & (3)\end{matrix}$ $\begin{matrix}{{{Az} + {By}} \leq \begin{matrix}d & ( {{all}{Secondary}{Distribution}{equation}} )\end{matrix}} & (4)\end{matrix}$ $\begin{matrix}{{x,y,{z \geq 0}}{x \in {{\mathbb{R}}^{n1}U\{ {0,1} \}^{p1}}}{z \in {{\mathbb{R}}^{n2}U\{ {0,1} \}^{p2}}}{y \in {\mathbb{R}}^{n3}}} & (5)\end{matrix}$

Where f_(Primary), g, and h represent the PIMS model; f_(Secondary) andthe constraints Az+By≤d represent the Secondary Distribution model;n1>>p1—Primary Distribution has significantly more continuous variablesthan discrete (tens of thousands vs. tens, or at most hundreds);n2>p2—Secondary Distribution has many more continuous than discretevariables (hundreds of thousands vs. thousands); n2>>n3,n1>>n3—Secondary Distribution and Primary Distribution share very fewvariables (tens or hundreds, at most); g and h are non-convex.

2. Customized Implementation of Benders to the Integrated Planning ofOil to Chemicals Decompositions includes the following steps (a)-(i):

Step (a): Partition the x variables in two: x^(S) (Stream Qualities),x^(M) (all other variables, mostly material flows). x^(S) are all thequalities (Sulfur, RON, etc.) that appear predominantly in nonlinearterms in blending equations. In the illustrative equationf1q1+f2q2−f3q3=0, the q variables belong to x^(S) and the f variablesbelong to x^(M).

Step (b): The complicating variables are y.

Step (c): Define a relaxed master problem (MR) by substituting nonlinearconstraints g and h for linearized relaxations.

$\begin{matrix}{{\max\limits_{x^{M},y,z}\pi}{s.t.}} & (6)\end{matrix}$ $\begin{matrix}{\pi \leq {{f_{Primary}( {x^{M},y} )} + {f_{Secondary}( {z,y} )}}} & (7)\end{matrix}$ $\begin{matrix}{{{Az} + {By}} \leq d} & (8)\end{matrix}$ $\begin{matrix}{{{Dx}^{M} + {Fy}} \leq 0} & (9)\end{matrix}$ $\begin{matrix}{{x^{M},y,{z \geq 0}}{x^{M} \in X^{M}}{z \in {{\mathbb{R}}^{n2}U\{ {0,1} \}^{p2}}}{y \in {\mathbb{R}}^{n3}}} & (10)\end{matrix}$

Step (d): Bilinear and trilinear terms of the type x_(i) ^(M)x_(j) ^(S)and x_(i) ^(M)x_(j) ^(S)x_(k) ^(S) constitute all or most of thenonlinearities in constraints g and h. An implementation linearizes thenonlinear constraints with these terms using a relaxed version ofMcCormick envelopes. The linearization strategies are described indetail in the next sections.

Step (e): The result of the linearization steps is that variables x^(S)are substituted by their upper and lower bounds, thus constraint setDx^(M)+Fy≤0 does not contain x^(S).

Step (f): Define the subproblem (S) for a fixed set of y^(k), z^(k).This is the original PIMS problem where the complicating variables y^(k)are fixed. The variables z^(k) are not complicating variables, but theycan be fixed from the solution of the master, as will be seen below.

$\begin{matrix}{{S = {\max\limits_{x^{M}}\{ {{f_{Primary}( {x^{M},y^{k}} )} + {f_{Secondary}( {z^{k},y^{k}} )}} \}}}{s.t.}} & (11)\end{matrix}$ $\begin{matrix}{{g( {x^{M},y^{k},x^{C},x^{S}} )} \leq 0} & (12)\end{matrix}$ $\begin{matrix}{{h( {x^{M},y^{k},x^{C},x^{S}} )} = 0} & (13)\end{matrix}$ $\begin{matrix}{{x^{M} \geq 0}{x^{s} \in X^{S}}{x^{M} \in X^{M}}} & (14)\end{matrix}$

Step (g): The solution of the relaxed master (MR) (a relaxed problemwithout the Benders cuts 16 b and 20) is a valid upper bound for theoriginal integrated problem (IP). The solution of the subproblem (S) isa lower bound for (IP). A Benders cut can be added to (MR) for everyoptimal solution of (S), of the form:

π ≤ f_(Primary)(x^(M^(k)), y) + f_(Secondary)(z, y) + μ^(k)g(x^(M^(k)), y, x^(S^(k))) + λ^(k)h(x^(M^(k)), y, x^(S^(k))).

Any y^(k) optimal in (MR) for which there is no solution in (S) rendersthe corresponding dual of (S) unbounded. Then, such an embodiment cutsthe infeasible y^(k) by applying a Benders feasibility cut

μ^(k)g(x^(M^(k)), y, x^(S^(k))) + λ^(k)h(x^(M^(k)), y, x^(S^(k))) ≤ 0.

Step (h): Define the Master Problem (M) as:

$\begin{matrix}{{\max\limits_{x^{M},y,z}\pi}{s.t.}} & (15)\end{matrix}$ $\begin{matrix}{\pi \leq {{f_{PIMS}( {x^{M},y} )} + {f_{SP}( {z,y} )}}} & ( {16a} )\end{matrix}$ $\begin{matrix}{{{\pi \leq {{f_{PIMS}( {x^{M^{k}},y} )} + {f_{SP}( {z,y} )} + {\mu^{k}{g( {x^{M^{k}},y,x^{S^{k}}} )}} + {\lambda^{k}{h( {x^{M^{k}},y,x^{S^{k}}} )}{for}k}}} = 1},\ldots,K^{Feas}} & ( {16b} )\end{matrix}$ $\begin{matrix}{{{Az} + {By}} \leq d} & (17)\end{matrix}$ $\begin{matrix}{{{Dx}^{M} + {Fy}} \leq 0} & (18)\end{matrix}$ $\begin{matrix}{{x^{M}y},{z \geq 0}} & (19)\end{matrix}$ $\begin{matrix}{{{{{{\mu^{k}{g( {x^{M^{k}},y,x^{S^{k}}} )}} + {\lambda^{k}{h( {x^{M^{k}},y,x^{S^{k}}} )}}} \leq {0{for}k}} = 1},\ldots,K^{Infeas}}{x^{M} \in X^{M}}{z \in {{\mathbb{R}}^{{n2} - {p2}}U\{ {0,1} \}^{p2}}}{y \in {\mathbb{R}}^{n3}}} & (20)\end{matrix}$

Step (i): Solve the integrated planning problem (IP) by iterativelysolving (M) and (S) and adding feasibility cuts to problem (M) until theoptimal solution of (M) and (S) are within a predetermined tolerance. Inan embodiment, the tolerance is defined as the percentage differencebetween the solution of (M) and (S). Typical values for this tolerance,according to an embodiment, are 0.5 to 5%.

Numerical Results with Real-Life Models

FIG. 6 is a simplified block diagram of a system 660 which may beoptimized using embodiments. The system 660 includes the productionnodes 661 a-c which correspond to respective real-world locations, e.g.,production plants. The nodes 661 a take raw materials 662 and, from theraw materials 662, generate products 663. Further, the system 660includes the finishing nodes 664 a-b (e.g., finishing plants), whichlikewise take raw materials 662 and, from the raw materials 662,generate products 663. Moreover, the finishing node 664 a performsfurther processes on output of the production node 661 c. In the system660, production node 661 b generates intermediates 667 which are sent todistribution node 668. It is noted that the node 668 may represent aplurality of distribution nodes, e.g., ten. Further, the products 663from the finishing nodes 664 a-b are sent to the distribution node 668which provides the products 663 to one or more points downstream, e.g.,customers. It should be understood that the nodes 661 a-c, 664 a-b, and668, may represent any locations in a supply chain for whichoptimization is desired. Further, embodiments are not limited to thestructure and relationships shown in FIG. 6 and other supply chains maybe optimized.

In an embodiment, the MINLP model that results from integrating the SPand PIMS models into one large-scale MINLP representing the system 660includes 482,077 variables (960 integers); 185,483 constraints (14,895non-linear); 2,179,608 non-zeros (91,972 non-linear). Further, for theexample results described below, there are 10 distribution nodes and 3time periods, corresponding to three months of operation.

For comparison, the aforementioned MINLP model was solved by“brute-force” using a MINLP solver, XLSP, provided by Applicant. XLSPdoes not guarantee global optimality. The results 770 of the“brute-force” solving are shown in FIG. 7. The “brute-force” results 770include the table 771 which has columns showing iterations 772, variableconvergence function 773, residual convergence function 774, objectiveconvergence function 775, and objective function value 776. The results770 also include table 777 that shows iterations 778 and the time 779for each iteration. The “brute-force” method determined a solution of€3,513 and took a total of 18 minutes and 15 seconds to determine thesolution.

FIG. 8 shows the results 880 of the same problem using the decompositionmethodology described therein. It is noted that the Master Problem andSubproblem are solved using the XSLP solver. The 20 iterations shown inthe table 881 are “internal” XSLP iterations used to solve the NLP. Thesolution of the problem only requires 1 and a half “External” Iterationsof the Decomposition algorithm (2 Master problems and 1 Subproblem),which are shown in the table 888. The results 880 include the table 881which has columns showing iterations 882, variable convergence function883, residual convergence function 884, objective convergence function885, objective function value 886, and non-linearity ratio 887. Theresults 880 further include the table 895 which shares the columnheadings 882-887 with table 881. The table 895 represents the results ofthe XSLP solver output after solving the second master problem. It canbe seen that the objective function value 886 in the table 895 matchesthe upper bound 892 of the table 888 discussed below. The table 895 is asingle line as opposed to multiple lines for the output from theSubproblem (table 881) because the master is a linear problem and thesubproblem is nonlinear (requiring multiple iterations).

The results 880 also include table 888 that shows iterations 889, type890, time 891, Upper Bound (Solution of the Master Problem) 892, LowerBound (Solution to the Subproblem) 893, and gap (difference between theupper and lower bound) 894. In the table 888, the values from iteration3 were derived utilizing “cuts” or additional equations derived from thesolution of the Subproblem in iteration 2 (row 2). The decompositionresult determined a solution of €3,532 and took a total of 7 minutes and10 seconds to reach the solution. As such, the decomposition methodologydetermined a better solution in less than half the time compared to thefull space solution.

External Inputs

Embodiments may utilize a variety of input data. Such input data mayinclude one or more models created for different nodes in the integratedsupply chain, e.g., the models 221 a-n. These models have been createdand ran using software applications, computer programs, or statisticaltools, for different nodes in the integrated supply chain. Such modelshave been used for planning, scheduling, and/or operation of each of thenodes.

Models of the supply, e.g., the models 221 a-n, include any kind ofrepresentation of a node in the supply chain that has an input-outputstructure where the inputs can be manipulated to optimize the outputswith respect to an objective function or performance indicator.Embodiments may utilize software applications, computer programs, orstatistical tools that are known to those of skill in the art togenerate those models.

Elements of Embodiments

Embodiments provide a methodology and computational tools to derive aset of mathematical constraints and equations using individual models ofeach node in a supply chain. Similarly, embodiments includemethodologies and computational tools to generate mathematicalstructure, e.g., linking structures 224 a-n to connect the differentmodels in the supply chain.

Embodiments also include software element(s) that can integrate thedifferent models and their connecting structures into a singlelarge-scale model, e.g., the model 225. Further, embodiments provide asoftware element that can optimally solve the large-scale modelmaximizing the value creation across the supply chain.

Embodiments implement an algorithmic strategy that exploits the businesscharacteristics (supply, demand, and product specifications), thephysical characteristics (material balances and other conservation laws)and the mathematical characteristics (linear and nonlinear) of thelarge-scale model to enable and expedite solving the large-scale model.In embodiments, the large-scale model is of such a size that it is notpossible to solve using existing technologies. Further still,embodiments implement an architectural set-up that allows the exchangeof information between the individual models used as input forembodiments and the integrated value chain optimization model.

Output of Embodiments

Embodiments determine optimal or close to optimal values for thevariables that are involved in the interface of different supply chainnodes. Such variables are often the driving force for integration ofnodes in supply chains, e.g., petroleum and chemical supply chains.Example outputs, e.g., 226, of embodiments include the optimalquantities of materials to be exchanged between an oil refinery and achemical plant belonging to the integrated supply chain. The output ofembodiments can take any form capable of communicating the determinedvalues or other such determinations made by embodiments describedherein. For example, embodiments can provide a data structure, file, orsignal that can pass the values of the variables back to the individualmodels used for each node in the supply chain.

Advantages Over Existing Software Applications

Existing software applications require that every node in the supplychain be modeled within the application, or at least within a suite ofapplications that are restricted by tight compatibility requirements.Embodiments have no such requirements and operate with any node modesthat have an input-output structure from which a mathematical model ofany kind can be extracted. This includes explicit algebraic equations orany black box model from which a data-driven mathematical model can beextracted.

Most existing software applications are limited to linear andmixed-linear optimization capabilities. Those applications that caninclude nonlinear elements are restricted to small problem sizes.Embodiments have the capabilities to solve very large-scaleMixed-Integer Nonlinear models.

Advantages Over Business Rules and Heuristic Methods

Business rules and heuristic methods are not based on rigorousanalytical and optimization principles. As such, business rules andheuristic methods almost universally fall short of maximal valuecreation across the supply chain. The exception being very trivialproblem instances. In contrast, embodiments provide optimized resultsfor controlling nodes of a supply chain.

Advantage Over Academic Mathematical Models

Academic models for determining optimal solutions of large-scale supplychain models do not contain enough detail of each supply chain node tobe of practical use. Moreover, academic models are not designed for thelay user and require post-graduate level expertise to set up and run.

Computer Support

FIG. 9 illustrates a computer network or similar digital processingenvironment in which embodiments of the present invention may beimplemented.

Client computer(s)/devices 50 and server computer(s) 60 provideprocessing, storage, and input/output devices executing applicationprograms and the like. Client computer(s)/devices 50 can also be linkedthrough communications network 70 to other computing devices, includingother client devices/processes 50 and server computer(s) 60.Communications network 70 can be part of a remote access network, aglobal network (e.g., the Internet), cloud computing servers or service,a worldwide collection of computers, Local area or Wide area networks,and gateways that currently use respective protocols (TCP/IP, Bluetooth,etc.) to communicate with one another. Other electronic device/computernetwork architectures are suitable.

FIG. 10 is a diagram of the internal structure of a computer (e.g.,client processor/device 50 or server computers 60) in the computersystem of FIG. 9. Each computer 50, 60 contains system bus 79, where abus is a set of hardware lines used for data transfer among thecomponents of a computer or processing system. Bus 79 is essentially ashared conduit that connects different elements of a computer system(e.g., processor, disk storage, memory, input/output ports, networkports, etc.) that enables the transfer of information between theelements. Attached to system bus 79 is I/O device interface 82 forconnecting various input and output devices (e.g., keyboard, mouse,displays, printers, speakers, etc.) to the computer 50, 60. Networkinterface 86 allows the computer to connect to various other devicesattached to a network (e.g., network 70 of FIG. 9). Memory 90 providesvolatile storage for computer software instructions 92 and data 94 usedto implement an embodiment of the present invention (e.g., the method100 of FIG. 1 detailed above). Disk storage 95 provides non-volatilestorage for computer software instructions 92 and data 94 used toimplement an embodiment of the present invention. The memory 90 and diskstorage 95 can be configured to store the various input-output models,equation oriented models, and optimization models described herein,along with the variables associated with said models, e.g., the linkingstructure and the interface variables. Moreover, the memory 90 and diskstorage 95 can be configured to implement the framework describedhereinabove in relation to FIG. 2 along with the associated models andvariables used to implement the framework. Central processor unit 84 isalso attached to system bus 79 and provides for the execution ofcomputer instructions.

In one embodiment, the processor routines 92 and data 94 are a computerprogram product (generally referenced 92), including a computer readablemedium (e.g., a removable storage medium such as one or more DVD-ROM's,CD-ROM's, diskettes, tapes, etc.) that provides at least a portion ofthe software instructions for the invention system. Computer programproduct 92 can be installed by any suitable software installationprocedure, as is well known in the art. In another embodiment, at leasta portion of the software instructions may also be downloaded over acable, communication and/or wireless connection. In other embodiments,the invention programs are a computer program propagated signal product107 embodied on a propagated signal on a propagation medium (e.g., aradio wave, an infrared wave, a laser wave, a sound wave, or anelectrical wave propagated over a global network such as the Internet,or other network(s)). Such carrier medium or signals provide at least aportion of the software instructions for the present inventionroutines/program 92.

In alternate embodiments, the propagated signal is an analog carrierwave or digital signal carried on the propagated medium. For example,the propagated signal may be a digitized signal propagated over a globalnetwork (e.g., the Internet), a telecommunications network, or othernetwork. In one embodiment, the propagated signal is a signal that istransmitted over the propagation medium over a period of time, such asthe instructions for a software application sent in packets over anetwork over a period of milliseconds, seconds, minutes, or longer. Inanother embodiment, the computer readable medium of computer programproduct 92 is a propagation medium that the computer system 50 mayreceive and read, such as by receiving the propagation medium andidentifying a propagated signal embodied in the propagation medium, asdescribed above for computer program propagated signal product.

Generally speaking, the term “carrier medium” or transient carrierencompasses the foregoing transient signals, propagated signals,propagated medium, storage medium and the like.

In other embodiments, the program product 92 may be implemented as a socalled Software as a Service (SaaS), or other installation orcommunication supporting end-users.

The teachings of all patents, published applications and referencescited herein are incorporated by reference in their entirety.

While example embodiments have been particularly shown and described, itwill be understood by those skilled in the art that various changes inform and details may be made therein without departing from the scope ofthe embodiments encompassed by the appended claims.

What is claimed is:
 1. A computer-implemented method for controlling anindustrial supply chain, the method comprising: for each node in asupply chain where the supply chain is formed of multiple nodes:obtaining an input-output model for the node; and generating anequation-oriented model using the obtained input-output modelcorresponding to the node; integrating the generated equation-orientedmodels of the multiple nodes with a linking structure to form anoptimization model of the supply chain, the optimization model of thesupply chain including a plurality of variables; using a categorizationof each of the plurality of variables, solving the optimization model ofthe supply chain to determine a value for at least one variable of theplurality; and outputting a signal indicating the determined value toenable controlling a given node of the multiple nodes in the supplychain in accordance with the determined value.
 2. The method of claim 1wherein each obtained input-output model includes one or more inputs andone or more outputs, wherein the one or more inputs are configured to bemanipulated to optimize the one or more outputs with respect to anobjective function or performance indicator.
 3. The method of claim 1wherein generating an equation-oriented model comprises at least one of:processing a given input-output model to generate a matrix indicatinglogistic, economic, and operational constraints for a supply chain nodecorresponding to the given input-output model; using a giveninput-output model and fitting parameters to a first-principlesengineering model; and processing a given input-output model using atleast one of statistics, machine learning, and artificial intelligence.4. The method of claim 1 wherein the linking structure includes at leastone of: mathematical variables and equations connecting the generatedequation-oriented models.
 5. The method of claim 1 wherein solving theoptimization model comprises: assigning each of the plurality variablesto one of a plurality of categories.
 6. The method of claim 5 whereinthe plurality of categories comprises: (i) variables connecting nodes inthe supply chain, (ii) variables in an objective function, (iii)variables in linear constraints in the optimization model, and (iv)variables exclusively in nonlinear constraints.
 7. The method of claim 1wherein solving the optimization model comprises: grouping linearconstraints in the optimization model; grouping nonlinear constraints inthe optimization model; removing the grouped nonlinear constraints fromthe optimization model by: for each nonlinear constraint, creating acorresponding linear constraint by substituting each variablecategorized as exclusively in the nonlinear constraint, with a lower andupper bound of the variable; and replacing each nonlinear constraint inthe optimization model with the created corresponding linear constraint;solving the optimization model with the removed nonlinear constraints todetermine optimal values for variables categorized as connecting supplychain nodes or variables that appear in an objective function; passingthe determined optimal values back to given equation-oriented modelsthat contain the nonlinear constraints; solving the givenequation-oriented models, wherein: if an equation-oriented model isfeasible the solving converges; and if an equation-oriented model isinfeasible, the solving does not converge and, in response, (i) creatinga cut constraint in terms of variables categorized as at least one of:connecting supply chain nodes and appearing in an objective function and(ii) adding the cut constraint to the optimization model; anditeratively solving the optimization model until solving eachequation-oriented model converges.
 8. The method of claim 1 furthercomprising: obtaining the outputted signal; determining a controlparameter of the given node using the determined value, from theobtained signal, in a given equation-oriented model indicating behaviorof the given node; and operating the given node in accordance with thedetermined control parameter.
 9. The method of claim 1 wherein nodes inthe supply chain include: a refinery node, a petrochemical plant node,and a polymer plant node.
 10. A computer system for controlling anindustrial supply chain, the system comprising: a processor; and amemory with computer code instructions stored thereon, the processor andthe memory, with the computer code instructions, being configured tocause the system to: for each node in a supply chain where the supplychain is formed of multiple nodes: obtain an input-output model for thenode; and generate an equation-oriented model using the obtainedinput-output model corresponding to the node; integrate the generatedequation-oriented models of the multiple nodes with a linking structureto form an optimization model of the supply chain, the optimizationmodel of the supply chain including a plurality of variables; using acategorization of each of the plurality of variables, solve theoptimization model of the supply chain to determine a value for at leastone variable of the plurality; and output a signal indicating thedetermined value to enable controlling a given node of the multiplenodes in the supply chain in accordance with the determined value. 11.The system of claim 10 wherein each obtained input-output model includesone or more inputs and one or more outputs, wherein the one or moreinputs are configured to be manipulated to optimize the one or moreoutputs with respect to an objective function or performance indicator.12. The system of claim 10 wherein, in generating an equation-orientedmodel, the processor and memory, with the computer code instructions arefurther configured to cause the system to perform at least one of:processing a given input-output model to generate a matrix indicatinglogistic, economic, and operational constraints for a supply chain nodecorresponding to the given input-output model; using a giveninput-output model and fitting parameters to a first-principlesengineering model; and processing a given input-output model using atleast one of statistics, machine learning, and artificial intelligence.13. The system of claim 10 wherein the linking structure includes atleast one of: mathematical variables and equations connecting thegenerated equation-oriented models.
 14. The system of claim 10 wherein,in solving the optimization model, the processor and the memory, withthe computer code instructions, are configured to cause the system to:assign each of the plurality variables to one of a plurality ofcategories.
 15. The system of claim 14 wherein the plurality ofcategories comprises: (i) variables connecting nodes in the supplychain, (ii) variables in an objective function, (iii) variables inlinear constraints in the optimization model, and (iv) variablesexclusively in nonlinear constraints.
 16. The system of claim 15wherein, in solving the optimization model, the processor and thememory, with the computer code instructions, are configured to cause thesystem to: group linear constraints in the optimization model; groupnonlinear constraints in the optimization model; remove the groupednonlinear constraints from the optimization model by: for each nonlinearconstraint, creating a corresponding linear constraint by substitutingeach variable categorized as exclusively in the nonlinear constraint,with a lower and upper bound of the variable; and replacing eachnonlinear constraint in the optimization model with the createdcorresponding linear constraint; solve the optimization model with theremoved nonlinear constraints to determine optimal values for variablescategorized as connecting supply chain nodes or variables that appear inan objective function; pass the determined optimal values back to givenequation-oriented models that contain the nonlinear constraints; solvethe given equation-oriented models, wherein: if an equation-orientedmodel is feasible the solving converges; and if an equation-orientedmodel is infeasible, the solving does not converge and, in response, (i)creating a cut constraint in terms of variables categorized as at leastone of: connecting supply chain nodes and appearing in an objectivefunction and (ii) adding the cut constraint to the optimization model;and iteratively solve the optimization model until solving eachequation-oriented model converges.
 17. The system of claim 10 wherein,the processor and the memory, with the computer code instructions, areconfigured to cause the system to: obtain the outputted signal;determine a control parameter of the given node using the determinedvalue, from the obtained signal, in a given equation-oriented modelindicating behavior of the given node; and operate the given node inaccordance with the determined control parameter.
 18. The system ofclaim 10 wherein nodes in the supply chain include: a refinery node, apetrochemical plant node, and a polymer plant node.
 19. A computerprogram product for controlling an industrial supply chain, the computerprogram product comprising: one or more non-transitory computer-readablestorage devices and program instructions stored on at least one of theone or more storage devices, the program instructions, when loaded andexecuted by a processor, cause an apparatus associated with theprocessor to: for each node in a supply chain where the supply chain isformed of multiple nodes: obtain an input-output model for the node; andgenerate an equation-oriented model using the obtained input-outputmodel corresponding to the node; integrate the generatedequation-oriented models of the multiple nodes with a linking structureto form an optimization model of the supply chain, the optimizationmodel of the supply chain including a plurality of variables; using acategorization of each of the plurality of variables, solve theoptimization model of the supply chain to determine a value for at leastone variable of the plurality; and output a signal indicating thedetermined value to enable controlling a given node of the multiplenodes in the supply chain in accordance with the determined value. 20.The computer program product of claim 19 wherein, in solving theoptimization model, the program instructions, when loaded and executedby the processor, cause the apparatus to: group linear constraints inthe optimization model; group nonlinear constraints in the optimizationmodel; remove the grouped nonlinear constraints from the optimizationmodel by: for each nonlinear constraint, creating a corresponding linearconstraint by substituting each variable categorized as exclusively inthe nonlinear constraint, with a lower and upper bound of the variable;and replacing each nonlinear constraint in the optimization model withthe created corresponding linear constraint; solve the optimizationmodel with the removed nonlinear constraints to determine optimal valuesfor variables categorized as connecting supply chain nodes or variablesthat appear in an objective function; pass the determined optimal valuesback to given equation-oriented models that contain the nonlinearconstraints; solve the given equation-oriented models, wherein: if anequation-oriented model is feasible the solving converges; and if anequation-oriented model is infeasible, the solving does not convergeand, in response, (i) creating a cut constraint in terms of variablescategorized as at least one of: connecting supply chain nodes andappearing in an objective function and (ii) adding the cut constraint tothe optimization model; and iteratively solve the optimization modeluntil solving each equation-oriented model converges.